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Unformatted text preview: M Research Group Vector Primer Prepared by the Electromagnetic Research Group Faculty Dennis Nyquist Edward Rothwell Leo Kempel Shanker Balasubramaniam March 23, 2004 M Research Group Scalar vs. Vector Quantities Defined by magnitude and direction Must define a set of laws analogous to those for scalar quantities Examples: Force, velocity, electric field, and magnetic field Completely specified by a single number along with a given dimensional unit Obeys usual algebraic laws (e.g. addition, subtraction, multiplication, and division) Examples: Mass and charge 1 P 2 P 12 12 d d r r = Note: Sometimes vectors are denoted by an arrow above the symbol, sometimes by a line, and sometimes with a bold font. We will use both for presentation clarity. M Research Group Unit Vectors and Vector Length Definition: The length of a vector, its magnitude, is designated as: A = A r bold normal font A A r r = a Definition: A unit vector has a length of unity: Note: Unit vectors point in the direction of positive increase for the corresponding coordinate . Definition: Product of a vector A and a scalar, , is a vector having the direction of the vector, but the magnitude is times the original length a A A P r r r = = Note: Any vector may be expressed as the product of its length with its unit vector. M Research Group A Comment on Notation This is what we will use Various notation conventions exist A A r r = = = A A ( ) ( ) ( ) ( ) ( ) ( ) , , r | , , r z , , | z , , z , y , x | z , y , x Cartesian Cylindrical Vectors Spherical x x 1 i a x = = = Unit vectors A r = A On the board In the notes Three most common coordinate systems ( ) z , y , x ( ) ( ) z , , z , , r = ( ) ( ) = , , r , , R Cartesian Cylindrical Spherical M Research Group...
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